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working paper
department
of economics
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
MEASUREMENT ERROR AND EARNINGS DYNAMICS:
Some Estimates from the PSID Validation Study
Jorn-Stef f en Pischke
Dec.
93
94-1
APR
5
1994
Measurement Error and Earnings Dynamics:
Some
Estimates
From
the
PSID
Validation Study
Jorn-Steffen Pischke
Massachusetts Institute of Technology
Department of Economics
Cambridge,
MA 02139-4307
November 1992
Revised:
December 1993
Measurement Error and Earnings Dynamics:
Some
Estimates
From
the
PSID Validation Study
Jbrn-Steffen Pischke
ABSTRACT
Previous empirical work on measurement error
in
survey earnings data has shown that the variance
of the measurement error is roughly constant over time,
and
it is
it is
negatively correlated with true earnings,
autocorrelated with previous measurement errors.
where the measurement error stems from underreporting of
white noise component. The model
implies that there will be
some
fits
This paper proposes a simple model
transitory earnings fluctuations and a
well to data from the
PSID Validation
Study.
While
it
bias in estimates of variances and autocovariances these biases are
of the same magnitude so that autocorrelations will be estimated roughly correctly.
Keywords: Panel
estimator.
data, reporting error,
income, covariance structure model,
minimum
distance
INTRODUCTION
1.
It is
widely recognized that measurement error
survey data and that errors-in-variables can lead
is
a pervasive
phenomenon
to inconsistent estimates in
models. In a linear regression context, attenuation bias"can be cured
available.
if
in
microeconomic
many econometric
suitable instruments are
Empirical studies trying to identify, for example, earnings or consumption processes
typically have to ignore
metric structure
(e.g.
measurement
error altogether or
assume
that
it
follows a particular para-
uncorrelated white noise). Absent any direct knowledge about the properties
of the measurement error process such assumptions are invariably necessary for identification. This
paper
tries to establish
some simple
relationships between
measurement error and the dynamics of
true earnings.
Systematic research on response error plays an important role in other social sciences (see
Sudman and Bradburn 1974)
but
little
work has been undertaken
properties of reporting error in earnings data.
The few
to assess the
magnitude and
studies that have tried to obtain validation
common
variables to assess the extent of measurement error directly
make
assumptions highly questionable. The earlier studies
have usually obtained two inde-
in this area
the
identifying
pendent observations on the same variable. Both these records were potentially error ridden. The
study by Mellow and Sider (1983)
employee-employer responses
results of separate
little
to
for the
insight on the direct nature of the
was undertaken by
They obtained
an example of
two
variables.
measurement
as the
team claims
They used matched
Such studies can be instructive but yield
error.
researchers at the Institute for Social Research at the University of Michigan.
payroll records at a cooperating firm and simultaneously administered the Panel
PSID
to
approach.
trying to obtain accurate validation records for typical labor market variables
Study of Income Dynamics (PSID) questionnaire
known
this
Current Population Survey (CPS) questions and compared the
wage regressions
The first study
is
Validation Study).
have obtained very
Due
to the
to careful
covered employees (the study
is
therefore
checking of the firm records the research
reliable validation information.
Their results are reported
in
Duncan and Hill (1985) and Duncan and Mathiowetz (1985). This study has an optimistic message:
%
more than 80
results
was
of the variability of earnings
measurement
that the
CPS
sets, like the
occasions for the purpose
to assess
used a match between the
CPS
by the
CPS
is
typical regressors
assumptions of the classical errors-in-variables-model.
have been matched
to administrative
records at various
Samuhel, and Triest (1986)
Little,
hot deck procedure to impute missing earnings values.
Bound and Krueger (1991)
measurement
It is
more
it is
CPS and
Social Security Administration records to study the
The CPS-SSA match has
error.
various advantages compared to the
representative of the whole population and contains
a two period panel referring to 1976 and 1977.
may
many more
The disadvantage
be erroneous as well and these earnings are top coded
maximum. Bound and Krueger
CPS,
seems correlated with many
part of their
to assess the error introduced
Security earnings
surement
The unfortunate
and Internal Revenue Service records
properties of
Moreover,
to actual signal.
survey data quality. David,
analyzed a similar match between the
PSIDVS.
due
error in earnings
in earnings equations, thus violating the
Census data
is
error, constructed
pointed out two
new
findings with the
observations.
is that
at the Social Security
CPS-SSA
match.
by subtracting social security earnings from earnings reported
negatively correlated with true earnings, a
measurement error. Secondly, measurement
error
phenomenon they
Social
refer to as
mean
Meain the
reverting
significantly positively autocorrelated in
is
two
adjacent periods. Both features defy the classical measurement error model and seriously complicate
the estimation of earnings dynamics.
These findings were corroborated on a second wave of the PSID Validation Study by Bound,
Brown, Duncan and Rogers (1989, in press). The second wave was collected four years later through
Bound et al.
a similar survey at the same firm.
in
also find
mean reversion and a positive autocorrelation
measurement error extending over a four year period.
framework
that the
to
summarize
the impact of
impact of measurement error
None of these studies shed
measurement error
be small.
to correct estimates of earnings
in analyses of the intertemporal allocation
measurement error have
and Card 1989) or consumption
a regression context and conclude
on the question of how
households or individuals over time. Generally, very
structure of the
in
in the level of earnings is likely to
direct light
dynamics as they are often used
In addition, they develop a nice theoretical
(e.g.
to
be made
restrictive
in studies
at least
on the dynamic
of life cycle labor supply
Hall and Mishkin 1982).
how measurement error relates to earnings over time.
assumptions
of resources by
I
use the
Abowd
this
we need to know
PSID
Validation Study
To improve on
In this paper
(e.g.
to address this issue.
The
fact that survey earnings are only available for
limits. Fortunately, the dataset contains validation earnings for
up
to six
two years imposes some
years thus providing a lot
of information on the true earnings process.
I
sample.
begin in section 2 with a description of the data set and some
I
also revisit the basic findings of
and show that they largely hold up
in
my
sample.
by
about the
et al. (in press)
Section 3 presents a dynamic model for both
results provides
underreport transitory earnings fluctuations. Quantitatively, this
error, the bulk is explained
statistics
Bound and Krueger (1991) and Bound
measurement error and earnings. The estimation
measurement
summary
is
some evidence
that individuals
only a small fraction of the
total
component. The following
a classical white noise
section discusses the impact of these results for the study of earnings dynamics, in particular the
estimation of permanent and transitory components of earnings. Section 5 concludes.
2.
DATA AND BASIC FACTS ABOUT MEASUREMENT ERROR
For the PSID Validation Study a survey questionnaire analogous
but shorter
firm.
was administered
to a
number of employees
At the same time, payroll records
for these
at
to the
one used for the PSID
an unnamed Detroit area manufacturing
employees were obtained from the
firm.
The
producers stress the cooperation of the firm involved which allowed an error free transcription of
the
company
records.
The PSIDVS
consists of two waves. In the
first
wave, 387 employees were interviewed
A second survey
1983. This survey included questions about annual earnings in 1981 and 1982.
was conducted
in
1987 which included questions about earnings for every year from 1981
these years are provided as well.
Validation earnings for
all
made
many
to reinterview as
participants
part of the panel.
the 1987 survey so that a total of
275 of the original 1983
Additional hourly workers were interviewed in
492 interviews could be coded for
that year.
Thus, both survey
years consist of a cross section larger than the overlap that creates the panel. Since
on measurement error for earnings with a one year
cross sections and the panel.
to 1986.
For the 1987 wave an attempt was
of the original participants as possible.
became eventually
in
recall
I
will refer to
I
will concentrate
them as the 1986 and 1982
1
Before proceeding
I
eliminated observations that contained assigned values or zeros in the
reported or validation earnings in the relevant years.
also eliminated four
I
more observations
because the coded validation earnings seem questionable and they deviate significantly from the
rest
of the sample.
Bound
work with
et al. (in press) also
a
sample where they remove some
outliers
and question the validity of two of the 1986 validation values. Details on the omitted observations
are given in the Appendix.
asserting that
in levels
measurement error
1
income variables thus
error.
and 2 provide some basic sample
section, respectively.
take logs of the
I
proportional. All of the qualitative results hold up for earnings
is
and additive measurement
Tables
is
In line with the literature,
Most striking about
the
statistics for the
sample
is
panel dataset and the 1986 cross
the high tenure, 19 years
on average. This
of course due to the sampling design, only employees continuously with the firm between 1981
and 1987 became part of the sample. The sample
general.
is
therefore also older than the labor force in
Sample members are relatively well educated, the proportion of women
is
very low. About
half of the panel are salaried employees. Earnings in this sample are higher than in representative
cross sections. Remarkably, validation and reported earnings do not differ
much
in either
mean
or
variance. This implies that "measurement error," calculated as the difference between reported and
validation earnings, has a
mean
quite large, on the other hand.
close zero.
Its
skewness ranges from
indicating only small asymmetries but fat
of the distribution can be found in
I
will use this definition of
The standard
tails in
deviation of the measurement error
-1.3 to 1.4
the distribution of the
et
necessitates
some within period
al.
(in press).
measurement
Bound and Krueger (1991) and Bound
measurement error
Identification of the
restrictions
the survey and validation variables.
Given
moments
error.
to
Plots
This implies that
made by Duncan and
Hill (1985)
of the survey measurement error
on the joint distribution of the measurement error
that
some gross
1
et al. (in press).
in the analysis that follows.
validation earnings are measured correctly, an assumption also
and Bound
and the kurtosis from 8
is
in
outliers in the validation variables have
been eliminated the assumption of correct validation earnings seems as good as any other.
Simple cross section regressions of the log measurement error so defined on demographics
did not reveal any particular pattern and the explanatory
results are therefore not reported here.
power of these regressions
is
low. These
Bound and Krueger (1991)
They found
the
that
established two important findings using the
measurement error is
two observations one year apart
measurement error and
Bound et
al.
positively correlated over time.
about 0.40. Secondly, there
is
true earnings in their data set.
(1989) on the PSIDVS. Bound et
for the error four years apart.
from the PSIDVS
I
will
show
al.
This
is
The
CPS-SSA
match.
autocorrelation between
a negative correlation between
has been confirmed by
latter finding
also found a positive but smaller autocorrelation
that these findings largely hold
data. In revisiting the earlier results
I
will
up on the samples
I
drew
pay particular attention to the distinction
between a recession year (1982) and a boom year (1986).
Table 3 reports the basic
results.
The
(m), true earnings (y), and reported earnings
%
and 25
stable
<Ty /<£
columns report
first
The variance of measurement
(x).
The variance of
of the variance of earnings.
error
is
between 15
measurement error seems much more
between 1982 and 1986 than the variance of earnings. This implies
that the reliability ratio
measurement error model
in a regression context,
,
which plays a crucial
will be higher in recessions.
also stressed by
Bound
role for the classical
However,
this is entirely related to the
variance of the signal, a finding
et al. (1989).
The last two columns report the regression
(b my )
the
measurement error
the variances of
and on reported earnings (b^). Bound
coefficients of measurement error on true earnings
et al. (in press)
show
that these regression coefficients
take the role of the reliability ratio for calculating the bias introduced by
general models; the reader
similar to theirs.
is
measurement error in more
referred to their paper for the algebraic details.
The negative correlation between measurement error and
for the recession year 1982 than for the
boom
year 1986. For the panel
My results are roughly
true earnings
hourly workers and the correlation
is
stronger
this coefficient is actually
estimated to be positive in 1986. However, a negative and significant coefficient
1986 cross section. This difference stems partly from the
is
is
found for the
fact that the cross section contains
more strongly negative
for hourly workers.
more
For the 1986 cross
section the coefficients are -0.170 (0.056) for hourly workers and 0.019 (0.035) for salaried
employees. For the panel the 1986 coefficients are -0.064 (0.067) for hourly workers and 0.020
(0.043) for the salaried group.
The
coefficient for the hourly workers differs to quite a substantial
degree between the cross section and the panel sample. For 1982 the panel coefficients are -0.105
(0.056) for hourly workers and -0.029 (0.052) for salaried employees.
holds up within the subgroups: the correlation
workers have always a lower correlation.
is
more negative
in
Thus
the general pattern
1982 for both of them. Hourly
The correlation between
same
pattern:
is
it
the
higher
measurement error and survey earnings
boom.
the
in
cov(m,x) = cov(m,y) + var(m)
This
has
Since the variance of
.
follow
to
m
is
positive but
by
definition
rather stable the
is
shows the
since
two covariance
terms have to follow the same relationship over time.
Table 3 reports results for the basic samples as well as for subsamples
true earnings to be
above 9 (about $8, 100)
fulfilled that restriction.
of the sample that
this
in
each year. For the 1986 cross section
There are a few observations with very low earnings
may strongly
influence the estimates involving
mainly reduces the estimate of the variance of y and x
estimates very much. All the results hold up
The
data
set.
is
slightly
lower than what Bound
y.
in the earlier years
As can be seen from
error
is
et al. (in press) found.
AR( 1)
Taking the Bound and Krueger (1991) estimate of 0.40
0.094 (0.065)
Let
me summarize
the three
structure of the
recessions. 2)
is
is
measurement
is
roughly constant over time,
stronger for hourly workers.
autocorrelated over time, the correlation
Since the
is
error.
possibility is that
combined with an AR(1).
Measurement error is weakly negatively correlated with
stronger in recessions and
in the panel
for the first order autocorrelation as a
main findings about measurement error
variance of the measurement error
the table,
They pointed out already
benchmark, the fourth order autocorrelation would be 0.026. One alternative
there exists a person specific fixed effect in misreporting
observations
samples as well.
measurement
that this correlation is too high to be compatible with an
all
1982 but changes none of the other
in
in the restricted
correlation between the 1986 and 1982
This
that restrict the log of
3)
it
may
at this stage:
1)
The
be slightly higher
in
true earnings, the correlation
Measurement
error
is
positively
declining slowly over time.
PSIDVS only contains two observations on measurement error I cannot distinguish
between different hypothesis on the correlation of the measurement error over time. However,
dataset contains relatively rich information on true earnings.
cations 1) and 2) reported above. In the next section,
between the dynamic structure of
true earnings
I
I
will therefore focus
the
on the impli-
use a simple model to capture the relationship
and measurement
error.
3.
In
A DYNAMIC MODEL FOR EARNINGS AND MEASUREMENT ERROR
many
applications
it is
useful to
decompose earnings
into a
permanent and a transitory
component. Different events are associated with these components. Promotions or job
high paying industry lead
to
permanent changes
induce temporary variations. Obviously,
the
more important events
it
is
in people's lives.
in
loss in a
earnings while overtime or temporary layoffs
the changes in
According
to
permanent earnings
that are related to
Sudman and Bradburn (1974)
events of
greater importance ("salience") are recalled better by survey respondents. This behavioral
suggests that respondents underreport transitory earnings changes. Such a model
with the findings reported above since transitory income fluctuations tend
A
simple
statistical
formulation of such a model
composed of a permanent (random walk) component and
V„
Z;«
In the estimation
Some
it is
I
chose
+
Z„-i
be larger in recessions.
Let annual earnings be
a transitory (white noise)
component,
i.e.
Tla
(1)
£,,
to let the transitory
random walk component
var(e
The process
=
+
Z it
as follows.
also consistent
not possible to identify time varying variances for both shocks for each period.
restriction is necessary;
the variance of the
=
is
to
is
model
for
it
)
=
a;,
is
var{y\
measurement error
is
m„ =
shock have time varying variances while
constant over time.
it
)
= aj„
cov(e,„ri li )
=
(2)
given by
ari,,
+
]i,
+
%„
var%) = (^
var((i,)
= a;
Measurement error consists of three components. The
shock,
we expect a <
.
The second component
is
first is
(3)
correlated with the transitory income
a person fixed effect in the
measurement error
unrelated to earnings. Recall that this will yield results that are consistent with the autocorrelation
in
measurement
error found by
Bound and Krueger
(1991). Given the limitations of the datasetat
hand no richer structure of the dynamics of the measurement error can be
is
just an arbitrarily
chosen alternative
measurementerror is white noise.
earnings at
model
to
In this simple
identified; the fixed effect
The
the autocorrelation.
introduced by varying
it is
the
model measurement error is not related to permanent
useful to difference the earnings equation
initial
conditions
=
( 1 )
to
eliminate heterogeneity
person effects in the earnings process).
(i.e.
Ay,,
+
e„
t]„
-
ri,.,
(4)
The resulting model for earnings and measurement error defined in equations
the following
(2), (3),
and
(4) implies
moment conditions
var(Ay
lt
cov(Ay,
/
var{m
lt
)
)
= <TE + 0^.+
,A.v,,_
=
l
(
_
+
<s\
)
= -acr,_,
= aj
restrict the
it
helps to put
sample so
some
(5)
;
All other sample covariances are restricted to be zero.
model. However,
+ aj
= acr,
cov(Ay,,,m,,_,)
cov(m, p m,
oj,.,
= -c^,.,
)
ara],
cflv(Ay,„/?7,.,)
I
component of
all.
For the estimation
that
last
This
is
structure on the data and
that the log of true earnings is
admittedly a very simple minded
it fits
above
9.
amazingly well. But
recall
Including observations with
smaller earnings often led to results that were strongly influenced by a single or a few observations,
especially in the estimates of the earnings process.
I fit
the
moment
conditions in (5) to the data by
matrix of earnings changes (for 1982
then imposing the restrictions using a
Abowd and Card
to
No
estimating the unrestricted covariance
1986) and measurement errors (for 1982 and 1986) and
minimum
1989). This strategy has a
Likelihood estimation.
first
distance estimator
(see
Chamberlain 1984 and
number of advantages compared
distributional assumptions are necessary.
to, say,
The estimation
Maximum
is
therefore
robust to the departures from normality typically observed in actual data. Simple specification tests
are available, allowing a general assessment of the
Given
1985).
can be traced
that the first stage unrestricted covariance matrix is available, failures of the
to their sources relatively easily.
estimates are asymptotically efficient.
putationally simpler than
small set of
Newey
of the model (Chamberlain 1984 and
fit
moments
The minimum
If
model
optimal weights are chosen the second stage
samples the two stage procedure
Finally, in large
com-
is
MLE because the nonlinear maximization has to be performed only on a
rather than the entire dataset.
distance estimator for a sample of size
7" is
given by
6 T = argmin{c T - f(b))'WT (cT - f(b))
where cT
WT
is
F =
is
a vector of r sample covariances,/(b)
is
a function of the parameters of interest, and
a positive semi-definite weighting matrix that possibly depends on the data.
df{b)ldb
,
the jacobian of the
moment
conditions.
If the
weighting matrix
the matrix of sampling covariances of the estimates of the sample
certain optimality characteristics and
tribution with degrees of
is
T times
freedom equal
moment
the inverse of the fourth
to r
-
the variances
The problem tends
to
is
the inverse of
then the estimator has
minimized has a %
their
be worse the
The
results
Instead,
I
like this.
This results from the correlation
sampling variances, which are calculated from the same
fatter the tails
of the empirical distribution (see Altonji and
I
experimented
is split
sampling variances are calculated from different portions of the
were extremely imprecise and highly dependent on
the particular
sample
data.
split used.
use a weighting matrix with the optimal weights on the main diagonal and zeros elsewhere.
This corresponds to weighted nonlinear least squares on the sample moments.
condition receives a weight that
moments
-dis-
minimum distance estimator tends to underfit
with the independently weighted estimator suggested by Altonji and Segal where the sample
their
2
matrix of earnings changes and measurement errors.
Segal 1993). In the sample the kurtosis of log wage changes ranges from 10 to 27.
and the moments and
Define
rank(F). In this case the optimal weighting matrix
on the main diagonal of a covariance model
between the second moments and
moments
the quadratic form being
In empirical applications the optimally weighted
data
(6)
is
allowed
for.
The asymptotic
is
inversely proportional to
This estimator seems
to yield
its
accuracy but no dependence between
very reasonable estimates.
distribution of the estimator in (6)
is
Each moment
^f{S T -b) -> N(0,(F'WFy F'W\r WF'(F'WF))
l
V =
Newey
(1985) has shown that an asymptotic %
T(c T
The
sample
the
at
a generalized inverse of
estimation results are given
hand there
is
no evidence
correct,
it
may
model
is
can be formed for
-statistic
this
model
as
- f(B T)YQr(cT - f(S T )) ~ X l rank{F)
PT =
is
(7)
E(cc')
2
Qt =
where Qj
l
- FT (FT 'WT FT )' FT'WT
l
I
QT
in
(8)
.
Table
4.
The
specification test reveals that in the small
model does not
that the
just reflect on the
observations purged of gross outliers.
WV
fit
the data. This
low power of the
The estimate of a
is
may
not
mean
that
with a sample of only 231
test
about -0.25, meaning that individuals
underreport transitory earnings by 25 %. Unlike most of the variances in the model, this coefficient
is
not very well determined.
error
is
The standard
deviation of the white noise
quite large, around 0. 10 in both years. There
periods. This
means
is
due
to the
this variance differs across
%
in
1982 and 5
%
in
1986.
More
also
seem
quite sensible.
than 80
They imply
% of the variation in earnings changes is due to the transitory component.
shocks are
measurement
%
of the
white noise component.
The estimates for the earnings process
and 90
no evidence that
in
that the underreporting of transitory earnings accounts only for a small part
of the total variance in measurement error, 9
variance
is
component
much more important
the standard deviation
is
in the recession
about twice as large
years 1982-83 than in the
in the recession.
that
between 75
Transitory earnings
boom
years 1985-86,
This explains part of the stronger
negative correlation between earnings and measurement error in a recession year like 1982 compared
to a
boom
year like 1986. This
is
the second major fact about
measurement error from
the previous
section.
Table 5 reports the actual covariance matrix used
fitted
covariance matrix from the model.
These
in the estimation
and table 6 displays the
tables provide useful information
on the short-
comings of the model. The model explains most of the variances and autocovariances of the earnings
10
process well;
it
overpredicts the autocovariances between 1983/84 and 1984/85. This stems from
the fact that a large value for the variance of the transitory earnings
the earnings variances in 1983 and 1984. This
This deviation
may be due to
at the
same
boom because
is
fit
permanent component is nonstationary
not everybody benefits from an upswing
predicts three non-zero covariances between earnings and
cov(Ay i82 ,m ia2 ), cov(Ay ia3 ,m lS2 ), cov{Ay i86 ,m l36 )
there
necessary to
time.
The model
(
is
the only parameter entering the autocovariance.
is
the fact that the variance of the
as well and high during the early years of a
component
one other non-zero covariance
and measurement error
in 1986.
relation exists with earnings in
that the
It fits all
).
later years
up
is
to 1986.
While
significant, a % -test for the joint hypothesis that all seven covariances
p- value of 0.43.
and the 1986 measurement error
may just
well.
rather peculiar since
2
be zero are indeed zero has a
them very
error
However,
model misses, between earnings changes
This correlation of -0.21
any of the
three of
measurement
in
1982
no similar cor-
this single correlation is
cov(A y„ _,, m„) which should
Thus, the correlation between 1982 earnings changes
be due
to
sampling variation.
The model has a number of other implications. Many sample members had negative transitory
earnings shocks in 1982 due to the recession. Thus, the measurement error should be positive on
average for that year. According to table
1
this is
indeed the case but the
mean of the measurement
error is very small (and not statistically different from zero).
An alternative test for the model can
a linear regression where
would
result in a
r\ u
downward
is
be designed by considering equation (3) as the basis for
replaced by current earnings. Obviously,
inconsistent estimate of
a
if
the
model was
true this
since earnings contains both a temporary
and a permanent component. However, a consistent estimate could be obtained by instrumenting
earnings with a variable that
movements
is
just related with transitory earnings
in earnings.
For the sample
at
hand, relatively stable manufacturing workers, changes in annual hours will
satisfy this criterion relatively well.
changes
in
changes but not with permanent
Between 1981 and 1982 much of these changes
are due to
temporary layoffs and unemployment while between 1985 and 1986 most will be due
to variation in overtime.
size significantly.
However, hours
are only available for hourly workers, cutting the
sample
Table 7 displays the results of regressing the measurement error on earnings
using hours as an instrument for the panel sample as well as the larger cross sectional samples.
coefficient estimates are similar to the ones obtained from the structural
11
The
model above except
for
—
.
the estimate for the 1986 cross section. This yields a point estimate of -0.54, indicating that for this
group one half of transitory earnings movements may go unreported.
some of the
It
should be kept
in
mind
that
difference in these estimates will be due to sampling variance.
We would
like to assess again
how much
the underreporting of transitory income,
Obviously, the standard
R
2
i.e.
of the variance of the measurement error
we
defined as explained
total
will notestimate this quantity since the variance oftheregressor, earnings, exceeds
estimator for the explained variance
available under the null that the
is
model
due
to
2
oro ,/^
like an estimate for the quantity
sum of squares divided by
is
sum of squares
a2 A consistent
,
.
is true.
given
It is
by
«,
.
R- =
—
covjm^y,,)
oc/v
var[m
This quantity
variation in
is
(9)
it
)
reported at the bottom of table 7.
measurement error
is
due
For the panel sample only 7
This
to underreporting.
is
surement error component.
better.
This will be true
more
In a
if the
still
from other factors
representative sample the
like the
higher.
Since
this is likely to
4.
One
in
white noise mea-
is
component in earnings
true.
IMPLICATIONS FOR THE ESTIMATION OF EARNINGS DYNAMICS
of the motivations for this paper
is to
We want to estimate the variances of the
to the transitory
i.e.
to three
earnings are due to mobility between firms and separations
assess to
what degree measurement error biases
estimates of the parameters in a dynamic model of earnings.
again.
two
variance of the white noise component of the measurement error
many movements
be
is
model might actually perform
the same in the population as in the sample while the variance of the transitory
is
of the
again similar to the finding above
using the structural estimates. In the cross section samples the explained variance
times as large but the major contribution comes
%
component,
r\ it
The only
.
data are only available on xu
Axu =
.
+
(1
in (1)
innovation to the permanent component, e„
data available are contaminated by
Instead of the true
£„
Consider the simple model
model
+oc)(r|„-ri„-
12
1
)
+
in (4)
kn
~
measurement
,
and
error,
we have
S* -i
(
10 )
The moment conditions
are
var(Axu ) = <Te +
+a)
(1
2
(aj,
+ aj,_
1
)
+ oj + a|_,
= -(l+a)o5,_, -
cov(Ax,,Ax,_,)
crj,.,
(11)
The estimates of the variance of true earnings will be confounded by two bias terms of opposite
sign. First, the variance will be underestimated
a<
because
On
.
the other hand, the white noise
term in measurement error % introduces an additional positive variance component. Similar biases
t
and 9 summarize these biases for 1982 and 1986 using
arise for the first autocovariance. Tables 8
the estimates
The
first
2
((1
from Table
+ oc) -
column
l)(c^,
4.
reports the bias due to underreporting of transitory income,
+ o^,_,)
measurement error.
Column
.
use 2o|
I
,
assuming
periods. Recall that this variance
this
component has substantial
changes (column
transitory
component.
This
to the
white noise component
that the variance of the noise is the
variance, in the
is
due
given by
same
in
in the
two adjacent
only estimated for the survey years 1982 and 1986. Notice that
is
However, the
4).
(2) gives the bias
it is
same order of magnitude
much lower because
total bias is
as the variance of earnings
of the underreporting of the
especially true for the recession year 1982 because of the large
variance of the transitory component.
The
first
panel uses actual period
t
and period
t-1
estimates of the variances in the transitory
earnings shock. This creates an asymmetry with the white noise error component.
panel in the table uses only period
the
same
in t-1.
t
The reliability
in
1982 because
Bound and Krueger(1991) found
reliability ratio is
very accurate as the
the entire
total
is
changes
to
of course
was
around
is
for 1976-77.
variance in
much
,
around one
economy while my
changes
this
in the bias
due
underreporting of the transitory
to
the highest variance year.
ratio for 1982, i.e. the ratio of variance of true earnings
of survey earnings changes oi/ci,
where the
estimates for both variables and assumes the variances were
This yields more extreme values
component especially
half.
The
This
picture
Bound
measurement
is
is
is
changes
much
mind
in
much
bleaker for the
that this
PSID
the 0.65 that
boom
The variance
than in the
year 1986
comparison
larger since their data
is
not
come from
in within plant earnings
et al. (in press) report the variance
in the
to the variance
somewhat higher than
One should keep
Bound and Krueger
be about three times as high
individual effects of the
0.8.
data just cover one plant.
lower.
The second
of reported earnings
PSIDVS. The white
noise and
error are presumably comparable in the population and the
13
single plant use here. Hence, there will be
error in a representative sample.
population,
mean
two sources of reduction
variance of
First, the
out that the strong business cycle pattern present in the plant level
the
PSIDVS
it
computed by using period
t
column 4
is
the
For the
data.
first
same
in the
reliability
worth pointing
is
not present in
in the population.
autocovariance biases can only be
innovation variances and assuming that the variances were the same
during the two previous years.
line of
PSIDVS
is
data
PSID. Thus, the differences between 1982 and 1986 should be much lower
Return to the analysis of the
the
is
Secondly, the
Furthermore,
is larger.
from measurement
a
will rise, so if
reversion will be stronger in a representative sample.
be higher because the variance of the signal
ratio will
r|„
in the biases
This yields the results given
in table 9.
one between 1983 and 1982. The biases
The covariance
in the first
in the first
autocovariance show a
very similar business cycle pattern as the variance.
Table 10 presents some
a permanent effect
which
is
(
aE and
results for the
decomposition of the variance of earnings changes into
an
a transitory effect
.
These are based on the second panel
consistent with the assumptions for the autocovariances in table
The
present the estimates for the standard deviations of the innovations.
The
validation earnings from table 4 are given below in parentheses.
identified
from the autocovariance. Given
from the variance. For 1982 the
its
for earnings: the first order autocorrelation
estimated.
As
estimate, the permanent
results are not
is
less than -0.5.
to the total variance of earnings
error
was purely white
noise.
the variance of the permanent
The
would therefore be
relative contributions
is
and
(3)
estimates based on the
transitory
component
component can be
is
identified
For 1986 both components are over-
changes
relatively
good news
is
given
in a similar
in
column
(4).
It
can be
way. This would not be true
the variance of the transitory
component
component would be estimated
accurately.
Then only
would be overstated while
This
(2)
compatible with a random walk plus noise model
seen that the measurement error biases both components
measurement
Columns
a measure of the relative importance of the two components the contribution of
permanent innovations
if
8.
in table 9
affected.
for the estimation of earnings dynamics: the negative correlation
of measurement error with transitory earnings attenuates the role of white noise measurement error.
The low reliability
ratios for this
sample should not be seen too pessimistically. The sample exhibits
much
less variation in the true variables than typical labor
many
different jobs, industries, and regions.
decomposition
is
market datasets that include people
in
Furthermore, the estimated permanent/transitory
roughly correct since the biases
in the
estimated variances and autocovariances
are of about equal magnitude. This result will only carry over to a
14
more representative sample
as
)
long as the transitory earnings variance does not
in further
rise too
reducing the impact of measurement error.
much.
A small rise will actually be helpful
If the transitory
variance becomes sufficiently
large the bias due to the underreporting feature will dominate the bias due to the white noise error
component.
This would imply that the importance of the transitory component might even be
understated in a permanent/transitory decomposition.
5.
CONCLUDING COMMENTS
This paper analyzes the structure of measurement error
of the true earnings process. The
PSID
Validation Study
in
earnings in relation to the dynamics
a fairly unique data set that allows such
is
an investigation. But the data also have important limitations. The small number of observations
in the data set make
it
hard to statistically discriminate between opposing hypotheses. Furthermore,
only two observations on survey earnings are available and these are not from adjacent years.
Finally, the data
come from
a single plant and
importantly, the variance in the responses
is
may
therefore not be very representative.
Most
also limited by this fact.
The results obtained here are qualitatively similar to
the findings of Bound
and Krueger ( 199 1
on the CPS-SSA match file. The PSIDVSdataexhibitmeanrevertingmeasurementerrorin earnings,
i.e.
the
measurement
lation is
weaker than
error
in
is
negatively correlated with validation earnings. However, this corre-
Bound and Krueger. The
results differ
year mainly because the variance of the earnings process
properties of the
measurement error changes
less cyclical than
manufacturing
To
characterize the data using a
measurement
boom and
a
a recession
different in these states while the
the extend that the
this finding will not easily carry
find a positive autocorrelation in the
I
little.
is
between
over
economy
as a
whole
to a larger population.
I
is
also
error over four years.
model where individuals underreport
the transitory
component
of earnings. This seems like a sensible behavioral hypothesis since permanent income changes are
often associated with
more important events and should
therefore be
remembered
better.
This part
of the measurement error implies that the transitory component in earnings will be underestimated
in a
model of earnings dynamics. However,
the white noise
component
in
measurement error more
than offsets this effect. This implies that the permanence in earnings changes
somewhat
if
inferences are
made on
the basis of data with
15
measurement
may
error. It
be understated
would be wrong,
however,
to attribute the entire transitory
better rule of
thumb
is
probably
component of earnings changes
to treat the
to
measurement error.
A
estimated transitory/permanent decompositions of
earnings as correct.
Overall, the
model
fits
the data well
when estimated
identified using an instrumental variables strategy.
sistent with the stylized facts about
Some
of survey and validation records would be necessary
component of earnings
On
model are not
features of the
fully con-
in misreporting.
A more extensive panel
to investigate this aspect in the data. In
representative samples, earnings dynamics also do not
also exhibit richer dynamics.
model or when
measurement error. The correlation of measurement errors over
time cannot be fully accounted for by a person fixed effect
as well. If the transitory
as a covariance structure
fit
is itself
the other hand, this
the
random walk
serially correlated
may
also call for a
plus white noise
more
model
measurement error
will
more elaborate model of
misreporting.
ACKNOWLEDGEMENTS
I
thank David Card, Francois Laisney, and participants
Mannheim Labor Lunches
for helpful
comments
at the
Princeton and University of
as well as an Associate Editor
for valuable suggestions.
16
and two referees
APPENDIX: OMITTED OBSERVATIONS
Here
I
provide some information on the four observations which
Each observation was omitted because
that the record is actually erroneous but
that these records
any
may
I
should
by
on the basis of the available information there is a possibility
sample
however, that the decision
rather arbitrarily, no particular cutoffs
in the
their 1987-ID.
PSIDVS
wave has
individual in the 1987
The
mean
be coded incorrectly. Since the four observations have a large influence on
stress,
Each individual
omitted from the sample.
the validation record is questionable. This does not
(least squares) statistics calculated with the
analysis.
I
to
I
find
saver to exclude them from the
it
exclude these particular observations
is
taken
were used.
that is part of the
1983 wave has a 1983-ID, similarly, each
a 1987-ID. All four outliers are part of the panel.
identify
I
them
four omitted observations have the IDs 20, 127, 191, and 307. Information
about validation and reported earnings
is
displayed in table
Al
for these individuals.
The ques-
tionable records are highlighted.
Observation 20
is
an hourly worker with 26 years of tenure earning an hourly
each year from 1982 to 1986. In 1981 he only worked 64 hours.
$10
in
that
someone keeps working
The other
them. They
all
It
seems inconceivable
for less than a dollar an hour even for such a short time.
three observations are salaried employees, no hours information
is
available for
have 16 or more years of education and between 12 and 18 years of tenure. In each
case the highlighted earnings reflect a significant drop in normal earnings levels.
report earnings
wage of about
much in
line with earnings in the other years. It
would not remember earnings changes of such a magnitude
seems peculiar
after
The respondents
that the respondents
one or two years. There are no
other college educated employees in the sample that had earnings changes even approaching the
same magnitude.
17
REFERENCES
Abowd,
and Card, D. (1989), "On the Covariance Structure of Earnings and Hours Changes,"
J.
Econometrica, 57, 41 1-445.
Altonji,
Brown, C, Duncan, G.
J.,
in
GMM
Estimation of Covariance
working paper. Department of Economics, Northwestern University.
Structures,"
Bound,
"Small Sample Bias
G. and Segal, L. (1993),
J.
J.,
and Rodgers,
W.
L. (1989),
"Measurement Error
sectional and Longitudinal Labor Market Surveys: Results from
Two
in Cross-
Validation Studies,"
Working Paper No. 2885, National Bureau of Economic Research
"Evidence on the Validity of Cross-Sectional and Longitudinal Labor Market
(in press),
Data," Journal of Labor Economics.
Bound,
J.
and Krueger, A. B. (1991), "The Extent of Measurement Error
Do Two Wrongs Make
Data:
David, M.,
Intriligator,
Little,
R.
J.
in
Handbook of Econometrics
J.
and
A., Samuhel,
Hill,
surement Error
Duncan, G.
J.
in
(Vol. 2), eds. Z. Griliches
Amsterdam: North Holland, 1247-1318.
M.
E.,
and Triest, R. E. (1986), "Alternative Methods for CPS
Income Imputation," Journal of the American
Duncan, G.
Longitudinal Earnings
a Right?" Journal of Labor Economics, 9, 1-24.
Chamberlain, G. (1984), "Panel Data,"
and M. D.
in
Statistical Association, 81, 29-41.
D. H. (1985), "An Investigation of the Extent and Consequences of Mea-
Labor-economic Survey Data," Journal of Labor Economics,
and Mathiowetz, N. A. (1985),
A
3,
508-532.
Validation Study of Economic Survey Data,
Ann
Arbor, MI: Institute for Social Research.
Hall, R. E. and Mishkin, F. S. (1982),
"The Sensitivity of Consumption
to
Transitory Income:
Estimates from Panel Data on Households," Econometrica, 50, 461-481.
Mellow, W. and Sider, H. (1983), "Accuracy of Response
Implications," Journal of Labor Economics,
1,
in
Labor Market Surveys: Evidence and
33 1-344.
Newey, W. K. (1985), "Generalized Method of Moments
Specification Testing,"
Journal of
Econometrics, 29, 229-256.
Sudman,
S.
and Bradburn, N. M. (1974), Response Effects
Chicago, IL: Aldine Publishing Compnay.
18
in Surveys.
A Review and
Synthesis,
Table
1.
Sample
Statistics:
Panel
Minimum
Maximum
0.29
8.71
10.96
10.28
0.30
8.58
11.00
Measurement Error 82
0.01
0.12
-0.48
0.62
Validation Earnings 86
10.47
0.21
9.76
11.02
Survey Earnings 86
10.47
0.24
9.62
10.97
Measurement Error 86
-0.00
0.11
-0.63
0.27
Highest Grade
13.0
1.94
8
17
Tenure
18.9
7.4
8
40
Age
40.3
8.1
29
63
Female
0.10
0.30
1
Salaried
0.50
0.50
1
Variable
Mean
Validation Earnings 82
10.27
Survey Earnings 82
Note:
Number of observations
is
Std.
234.
19
Dev.
Table
2.
Sample
Statistics:
1986 Cross Section
Variable
Mean
Validation Earnings 86
10.46
Survey Earnings 86
Minimum
Maximum
0.21
9.74
11.02
10.46
0.23
9.62
11.08
Measurement Error 86
-0.00
0.12
-0.63
0.55
Highest Grade
12.58
2.07
4
17
Tenure
20.30
8.03
8
43
Age
41.67
8.67
27
63
Female
0.08
0.27
1
Salaried
0.33
0.47
1
Note:
Number of observations
is
Std.
437.
20
Dev.
Table
Data
3.
Basic Facts About Measurement Error in Earnings
set
Panel 1986
Panel 1986 (y > 9)
Cross Sect. 1986
Panel 1982
Panel 1982 (y > 9)
Cross Sect. 1982
Cross Sect. 1982
nobs
var(m)
var(y)
var(;t)
Ky
b„ix
234
0.0110
0.0441
0.0557
0.007
0.203
(0.037)
(0.032)
0.006
0.201
(0.037)
(0.032)
-0.068
0.207
(0.036)
(0.026)
231
437
234
231
352
351
0.0110
0.0446
0.0440
0.0137
0.0135
0.0860
0.0697
0.0135
0.0142
0.0739
0.0670
0.0142
0.0562
0.0517
0.0875
0.0697
0.0735
0.0654
(y>9)
Note: Heteroskedasticity consistent standard errors
21
in parentheses.
-0.070
0.086
(0.038)
(0.026)
-0.097
0.097
(0.042)
(0.031)
-0.099
0.094
(0.035)
(0.023)
-0.118
0.096
(0.034)
(0.025)
Table
4.
Minimum
Parameter
Estimate
Std. Err.
*n8i
0.106
0.027
0.139
0.022
0.132
0.021
>Tl84
0.120
0.033
>Tl85
0.073
0.021
J
0.089
0.025
J
J
2
X
Distance Estimates for Earnings Changes and Measurement Error
Tl82
n83
tl86
Parameter
Std. Err.
0.042
0.055
0.105
0.011
0.096
0.010
a,
0.034
0.013
a
-0.258
0.123
^82
23.870 [17]
-Statistic [dof]
0.123
p-value
Note: Panel data, number of observations
Estimate
is
23
1.
22
Table
5.
Estimated Covariance Matrix of Earnings and Measurement Error
Ay«2
Ayiu
Ay,w
Ay«5
Ay i86
fn-,82
"li86
tym
0.0322
-0.54
-0.05
-0.18
0.09
-0.21
-0.21
Ay«i
-0.0193
0.0399
-0.45
0.17
0.07
0.22
0.08
Ay,«
-0.0018
-0.0163
0.0335
-0.43
-0.06
0.05
0.01
Ay«5
-0.0049
0.0051
-0.0121
0.0239
-0.30
0.09
0.00
Ayi85
0.0019
0.0016
-0.0012
-0.0055
0.0141
-0.05
-0.18
™\82
-0.0044
0.0051
0.0011
0.0016
-0.0007
0.0135
0.10
™i86
-0.0039
0.0016
0.0001
0.0000
-0.0023
0.0012
0.0110
Note: Covariances are below the diagonal, correlations above the diagonal
23
Table
6.
Fitted Covariance Matrix
Ay lS2
Ay aj
Ay lS4
Ay iS5
Ay i86
mm
m,i86
Ay i82
0.0322
-0.55
0.00
0.00
0.00
-0.24
0.00
Ayi83
-0.0192
0.0383
-0.48
0.00
0.00
0.22
0.00
Ayi84
0.0000
-0.0173
0.0334
-0.54
0.00
0.00
0.00
Ay l85
0.0000
0.0000
-0.0144
0.0214
-0.30
0.00
0.00
Ayi86
0.0000
0.0000
0.0000
-0.0053
0.0149
0.00
-0.16
m
i82
-0.0050
0.0050
0.0000
0.0000
0.0000
0.0135
0.10
m
i86
0.0000
0.0000
0.0000
0.0000
-0.0020
0.0012
0.0110
Note: Covariances are below the diagonal, correlations above the diagonal
24
Table
7.
Regression of Measurement Error on Earnings Using Hours Changes as Instruments
1986 Panel
Coefficient
R2
no. of obs.
1982 Panel
R
1982 Cross
Section
Section
-0.31
-0.15
-0.23
-0.54
(0.20)
(0.12)
(0.93)
(0.17)
0.07
0.07
0.14
0.23
114
114
168
292
Note: Heteroskedasticity consistent standard errors
the reported
1986 Cross
2
.
25
in parentheses.
See
text for the definition of
Table
Year
8.
The Biases
in
Estimated Variances of Earnings
Underrep. of
White noise
trans, earnings
Error
(1)
(2)
Total Bias
(l)
+
var(Av„)
Reliability
Ratio
(2)
(3)
(4)
(5)
Actual Innovation Variances
1982
-0.0137
0.0221
0.0084
0.0322
0.79
1986
-0.0060
0.0184
0.0124
0.0141
0.53
Period
t
Innovation Variances
1982
-0.0174
0.0221
0.0047
0.0322
0.87
1986
-0.0071
0.0184
0.0113
0.0141
0.56
26
Table
Year
9.
The Biases
in the
Estimated First Order Autocovariance of Earnings
Underrep. of
trans,
earnings
White noise
Total Bias
(D +
EiTor
cov(Ay ,Ayt-i)
t
Reliability
Ratio
(2)
(1)
(2)
(3)
(4)
(5)
1982
0.0087
-0.0110
-0.0023
-0.0193
0.89
1986
0.0036
-0.0092
-0.0056
-0.0055
0.50
27
Table
Year
10.
Implied Estimates of the Innovation Variances
Gix,
Oa
<**
Contrib. of
Perm. Comp.
1982
1986
(1)
(2)
(3)
0.19
0.15
...
(0.18)
(0.14)
(0.04)
0.16
0.11
0.06
(0.12)
(0.09)
(0.04)
Note: Estimates from Table 4
in parentheses.
28
(4)
(5.5
12.6
%)
%
(12.5%)
Table Al. Earnings for Observations Omitted form the Final Sample
ID
Record
1981
1982
1983
1984
1985
1986
20
Validation
62
24204
17525
20379
33020
32224
Reported 83
2400
24000
Reported 87
4000
24000
23000
30000
32000
32000
Validation
29379
51951
55720
57258
57045
54195
Reported 83
52000
52000
Reported 87
50000
50000
55000
55000
55000
60000
Validation
5237
30653
37565
41557
40803
45327
Reported 83
26500
36000
Reported 87
45000
51000
52000
55000
62000
63500
Validation
40730
39691
44020
48307
52063
11076
Reported 87
40000
42000
43000
49000
51000
55000
127
191
307
29
026
MIT LIBRARIES
3
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